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diff --git a/theories/Relations/Newman.v b/theories/Relations/Newman.v new file mode 100755 index 00000000..3cf604d8 --- /dev/null +++ b/theories/Relations/Newman.v @@ -0,0 +1,123 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Newman.v,v 1.7.2.1 2004/07/16 19:31:16 herbelin Exp $ i*) + +Require Import Rstar. + +Section Newman. + +Variable A : Type. +Variable R : A -> A -> Prop. + +Let Rstar := Rstar A R. +Let Rstar_reflexive := Rstar_reflexive A R. +Let Rstar_transitive := Rstar_transitive A R. +Let Rstar_Rstar' := Rstar_Rstar' A R. + +Definition coherence (x y:A) := ex2 (Rstar x) (Rstar y). + +Theorem coherence_intro : + forall x y z:A, Rstar x z -> Rstar y z -> coherence x y. +Proof + fun (x y z:A) (h1:Rstar x z) (h2:Rstar y z) => + ex_intro2 (Rstar x) (Rstar y) z h1 h2. + +(** A very simple case of coherence : *) + +Lemma Rstar_coherence : forall x y:A, Rstar x y -> coherence x y. + Proof + fun (x y:A) (h:Rstar x y) => coherence_intro x y y h (Rstar_reflexive y). + +(** coherence is symmetric *) +Lemma coherence_sym : forall x y:A, coherence x y -> coherence y x. + Proof + fun (x y:A) (h:coherence x y) => + ex2_ind + (fun (w:A) (h1:Rstar x w) (h2:Rstar y w) => + coherence_intro y x w h2 h1) h. + +Definition confluence (x:A) := + forall y z:A, Rstar x y -> Rstar x z -> coherence y z. + +Definition local_confluence (x:A) := + forall y z:A, R x y -> R x z -> coherence y z. + +Definition noetherian := + forall (x:A) (P:A -> Prop), + (forall y:A, (forall z:A, R y z -> P z) -> P y) -> P x. + +Section Newman_section. + +(** The general hypotheses of the theorem *) + +Hypothesis Hyp1 : noetherian. +Hypothesis Hyp2 : forall x:A, local_confluence x. + +(** The induction hypothesis *) + +Section Induct. + Variable x : A. + Hypothesis hyp_ind : forall u:A, R x u -> confluence u. + +(** Confluence in [x] *) + + Variables y z : A. + Hypothesis h1 : Rstar x y. + Hypothesis h2 : Rstar x z. + +(** particular case [x->u] and [u->*y] *) +Section Newman_. + Variable u : A. + Hypothesis t1 : R x u. + Hypothesis t2 : Rstar u y. + +(** In the usual diagram, we assume also [x->v] and [v->*z] *) + +Theorem Diagram : forall (v:A) (u1:R x v) (u2:Rstar v z), coherence y z. + +Proof + (* We draw the diagram ! *) + fun (v:A) (u1:R x v) (u2:Rstar v z) => + ex2_ind + (* local confluence in x for u,v *) + (* gives w, u->*w and v->*w *) + (fun (w:A) (s1:Rstar u w) (s2:Rstar v w) => + ex2_ind + (* confluence in u => coherence(y,w) *) + (* gives a, y->*a and z->*a *) + (fun (a:A) (v1:Rstar y a) (v2:Rstar w a) => + ex2_ind + (* confluence in v => coherence(a,z) *) + (* gives b, a->*b and z->*b *) + (fun (b:A) (w1:Rstar a b) (w2:Rstar z b) => + coherence_intro y z b (Rstar_transitive y a b v1 w1) w2) + (hyp_ind v u1 a z (Rstar_transitive v w a s2 v2) u2)) + (hyp_ind u t1 y w t2 s1)) (Hyp2 x u v t1 u1). + +Theorem caseRxy : coherence y z. +Proof + Rstar_Rstar' x z h2 (fun v w:A => coherence y w) + (coherence_sym x y (Rstar_coherence x y h1)) (*i case x=z i*) + Diagram. (*i case x->v->*z i*) +End Newman_. + +Theorem Ind_proof : coherence y z. +Proof + Rstar_Rstar' x y h1 (fun u v:A => coherence v z) + (Rstar_coherence x z h2) (*i case x=y i*) + caseRxy. (*i case x->u->*z i*) +End Induct. + +Theorem Newman : forall x:A, confluence x. +Proof fun x:A => Hyp1 x confluence Ind_proof. + +End Newman_section. + + +End Newman. |